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Determinantendarstellung von Übergangsmatrixelementen für das Spin-½ XXZ-Modell

Daniel Biegel

Published Nov 10, 2003• 0 citations
OpenAlex

Abstract

The one-dimensional spin-½ Heisenberg model was solved, in principle, by H. Bethe in 1931 using an ad hoc trial wavefunction, now famously known as the Bethe ansatz. Decades went by before the power and scope of this method of exact analysis became widely known for applications to spectrum and thermodynamics of a select class of completely integrable model systems. Until recently, the most vexing exception to immense progress in the further development of the Bethe ansatz has been the absence of a practical method to use the exactly known and readily available Bethe wave functions for the explicit calculation of transition matrix elements. The knowledge of such transition rates is of paramount importance for an understanding of dynamic correlation functions in relation to the underlying quasiparticles and for the interpretation of experimental probes of quantum fluctuations in quasi-one-dimensional magnetic compounds. It was most remarkable, therefore, when Kitanine, Maillet, and Terras succeeded in reducing matrix elements between Bethe wave functions for local spin operators to determinantal expressions. Here these expressions are used to calculate dynamic spin structure factors of the Heisenberg antiferromagnet (XXX model) and the one-dimensional spin-½ XXZ model in the planar regime (|Δ| < 1), both with periodic boundary conditions and an external magnetic field: Determinantal representations are derived for the zero-temperature spin fluctuations parallel and perpendicular to the external field. The singular nature of the Bethe ansatz equations for the case Δ = 0 (XX model) is discussed and the XX limit is performed in the determinantal expressions for the transition matrix elements. As an application, lineshapes for dynamic structure factors relevant for experiments are calculated. The predominant excitations are identified within the framework of quasiparticles excited from a corresponding vacuum.

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Transition rates via Bethe ansatz for the spin-½ Heisenberg chain

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We use the exact determinantal representation derived by Kitanine, Maillet, and Terras for matrix elements of local spin operators between Bethe wave functions of the one-dimensional s = ½ Heisenberg model to calculate and numerically evaluate transition rates pertaining to dynamic spin structure factors. For real solutions z 1 ,..., z r of the Bethe ansatz equations, the size of the determinants is of order r × r . We present applications to the zero-temperature spin fluctuations parallel and perpendicular to an external magnetic field.

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Integrable models are physical models for which some quantities can be exactly obtained, without use of perturbation theory. Those very special models are source of an increasing interest in theoretical physics. The direct applications in condensed matter physics and the subtle links evidenced more recently with some supersymmetric gauges theories motivated the development of complex mathematical tools. Among these, Bethe ansatz played an important role, and provides an efficient approach for diagonalizing a lot of models of various nature. The first chapter of this thesis is devoted to the introduction to the two approaches of the Bethe ansatz, said “coordinate” and “algebraic”, in the context of the XXX Heisenberg spin chain and a continuous spin generalization of the Totally Asymmetric Simple Exclusion Process, the so called Zero-range Chipping model with factorized steady state (ZCM). The second chapter is devoted to the Modified Algebraic Bethe Ansatz in the context of the periodic XXX chain. This modified ansatz is proposed for solving the spectral problem of the open spin chain, for which the usual ansatz fails. The scalar product of the obtained modified Bethe states is studied. The third chapter concerns the resolution of the identity and the inverse functional problem. An expression for the spin states in terms of Bethe states est presented for the ZCM, and an expression for the resolution of the identity in term of Bethe states for the infinite XXZ chain is proved, involving in both cases the contribution of bound states. At last, the fourth chapter concerns determinant representations in the Bethe ansatz. An expression for the “matrix elements of the particle number operator” for the delta-Bose gas in terms of a determinant is proved, and some integral representations for the Izergin-Korepin and Slavnov determinants are investigated, then establishing a new formal link between these two determinant representations.

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The one-dimensional spin-½ Heisenberg model was solved, in principle, by H. Bethe in 1931 using
an ad hoc trial wavefunction, now famously known as the Bethe ansatz. Decades went by before the
power and scope of this method of exact analysis became widely known for applications to spectrum
and thermodynamics of a select class of completely integrable model systems.
Until recently, the most vexing exception to immense progress in the further development of the Bethe
ansatz has been the absence of a practical method to use the exactly known and readily available
Bethe wave functions for the explicit calculation of transition matrix elements. The knowledge of such
transition rates is of paramount importance for an understanding of dynamic correlation functions in
relation to the underlying quasiparticles and for the interpretation of experimental probes of quantum
fluctuations in quasi-one-dimensional magnetic compounds.
It was most remarkable, therefore, when Kitanine, Maillet, and Terras succeeded in reducing matrix
elements between Bethe wave functions for local spin operators to determinantal expressions.
Here these expressions are used to calculate dynamic spin structure factors of the Heisenberg antiferromagnet
(XXX model) and the one-dimensional spin-½ XXZ model in the planar regime
(|Δ| < 1), both with periodic boundary conditions and an external magnetic field:
Determinantal representations are derived for the zero-temperature spin
fluctuations parallel and perpendicular to the external field.
The singular nature of the Bethe ansatz equations for the case Δ =
0 (XX model) is discussed and the XX limit is performed in the determinantal expressions for the transition matrix elements.
As an application, lineshapes for dynamic structure factors relevant for experiments are calculated.
The predominant excitations are identified within the framework of quasiparticles excited
from a corresponding vacuum.
We use the exact determinantal representation derived by Kitanine, Maillet, and Terras for matrix elements of local spin operators between Bethe wave functions of the one-dimensional s = ½ Heisenberg model to calculate and numerically evaluate transition rates pertaining to dynamic spin structure factors. For real solutions z 1 ,..., z r of the Bethe ansatz equations, the size of the determinants is of order r × r . We present applications to the zero-temperature spin fluctuations parallel and perpendicular to an external magnetic field.
As part of a study that investigates the dynamics of the s=12s=\frac{1}{2}s=21​ XXZ\mathrm{XXZ}XXZ model in the planar regime $|\ensuremath{\Delta}|<1,$ we discuss the singular nature of the Bethe ansatz equations for the case $\ensuremath{\Delta}=0$ (XX(XX(XX model). We identify the general structure of the Bethe ansatz solutions for the entire XX\mathrm{XX}XX spectrum, which include states with real and complex magnon momenta. We discuss the relation between the spinon or magnon quasiparticles (Bethe ansatz) and the lattice fermions (Jordan-Wigner representation). We present determinantal expressions for transition rates of spin-fluctuation operators between Bethe wave functions and reduce them to product expressions. We apply the formulas to two-spinon transition rates for chains with up to N=4096N=4096N=4096 sites.
The spin fluctuations parallel to the external magnetic field in the ground state of the one-dimensional (1D) s=12s=\frac{1}{2}s=21​ Heisenberg antiferromagnet are dominated by a two-parameter set of collective excitations. In a cyclic chain of N sites and magnetization 0<Mz<N/2,0<{M}_{z}<N/2,0<Mz​<N/2, the ground state, which contains 2Mz{2M}_{z}2Mz​ spinons, is reconfigured as the physical vacuum for a different species of quasiparticles, identifiable in the framework of the coordinate Bethe ansatz by characteristic configurations of Bethe quantum numbers. The dynamically dominant excitations are found to be scattering states of two such quasiparticles. For $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty},$ these collective excitations form a continuum in $(q,\ensuremath{\omega})$ space with an incommensurate soft mode. Their matrix elements in the dynamic spin structure factor ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ are calculated directly from the Bethe wave functions for finite N. The resulting line- shape predictions for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ complement the exact results previously derived via algebraic analysis for the exact two-spinon part of ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ in the zero-field limit. They are relevant for neutron-scattering experiments on quasi-1D antiferromagnetic compounds in a strong magnetic field.
Integrable models are physical models for which some quantities can be exactly obtained, without use of perturbation theory. Those very special models are source of an increasing interest in theoretical physics. The direct applications in condensed matter physics and the subtle links evidenced more recently with some supersymmetric gauges theories motivated the development of complex mathematical tools. Among these, Bethe ansatz played an important role, and provides an efficient approach for diagonalizing a lot of models of various nature. The first chapter of this thesis is devoted to the introduction to the two approaches of the Bethe ansatz, said “coordinate” and “algebraic”, in the context of the XXX Heisenberg spin chain and a continuous spin generalization of the Totally Asymmetric Simple Exclusion Process, the so called Zero-range Chipping model with factorized steady state (ZCM). The second chapter is devoted to the Modified Algebraic Bethe Ansatz in the context of the periodic XXX chain. This modified ansatz is proposed for solving the spectral problem of the open spin chain, for which the usual ansatz fails. The scalar product of the obtained modified Bethe states is studied. The third chapter concerns the resolution of the identity and the inverse functional problem. An expression for the spin states in terms of Bethe states est presented for the ZCM, and an expression for the resolution of the identity in term of Bethe states for the infinite XXZ chain is proved, involving in both cases the contribution of bound states. At last, the fourth chapter concerns determinant representations in the Bethe ansatz. An expression for the “matrix elements of the particle number operator” for the delta-Bose gas in terms of a determinant is proved, and some integral representations for the Izergin-Korepin and Slavnov determinants are investigated, then establishing a new formal link between these two determinant representations.
Inelastic neutron scattering experiments are commonly used to unveil how excitations on Heisenberg spin models play a role in dynamical correlations functions. For a certain class of materials, like CsCoCl3_33​ or CsCoBr3_33​ salts, it turns out that their magnetic properties are fairly well approximated by quasi-one dimensional XXZ models, which enjoy the property of quantum integrability. In these instances, one can in principle use their underlying algebraic structure to describe very precisely how excitations, the so-called spinons, participate in dynamical correlations functions. Even though the available theories (either algebraic Bethe ansatz or quantum group approach) provide all the needed physical quantities such as form factors, complete set of eigenstates and spectrum, it is typically a rather daunting task, however, to obtain sufficiently simple analytical expressions for computing Dynamical Structure Factors (DSFs), valuable, e.g., for parameters estimation based on experimental data. This is particularly the case for the longitudinal DSF of the XXZ model, which has eluded a formal mathematical treatment thus far. Using the quantum group approach, we present here an exact and simple expression of the 2-spinon longitudinal DSF and show our results to be consistent with the expected sum rules and the isotropic and Ising antiferromagnet limiting cases.